Using Contracts to Influence the Outcome of a Game∗
Robert McGrew and Yoav Shoham
Computer Science Department
Stanford University
Stanford, CA 94305
{bmcgrew, shoham}@cs.stanford.edu
Abstract
We consider how much influence a center can exert on a game
if its only power is to propose contracts to the agents before
the original game, and enforce the contracts after the game
if all agents sign it. Modelling the situation as an extensiveform game, we note that the outcomes that are enforceable are
precisely those in which the payoff to each agent is higher
than its payoff in at least one of the Nash equilibria of the
original game. We then show that these outcomes can still be
achieved without any effort actually expended by the center:
We propose a mechanism in which the center does not monitor the game, and the contracts are written so that in equilibrium all agents sign and obey the contract, with no need for
center intervention.
Introduction
There has been much interest in AI in mechanism design, the
area of game theory devoted to designing protocols for selfinterested agents. In the literature (Mas-Colell, Whinston,
& Green 1995) it is generally assumed that the mechanism
designer has complete freedom in designing the rules of the
game. Yet the world is full of strategic situations with rules
that already exist and cannot be changed arbitrarily. Recent
work on k-implementation (Monderer & Tennenholtz 2003)
restricts the capabilities of the mechanism designer in a particular way – it can add to any given cell in the payoff matrix, but it cannot subtract. (The interesting results in that
line of work concern cases in which, despite that addition,
the cost to the center in equilibrium is zero.) The opposite
of this setting would be one in which the center can impose
fines, rather than bonuses. This in and of itself is not interesting, because with sufficiently large fines any outcome
can be enforced. However, suppose the mechanism cannot
unilaterally impose fines, but it can do so in the context of
a signed contract. Specifically, we consider the following
class of mechanisms. Given a game G, the center can:
1. Propose a contract before G is played. This contract specifies a particular outcome, that is, a unique action for each
agent, and a penalty for deviating from it.
∗
This work was supported in part by the National Science Foundation under ITR IIS-0205633.
c 2004, American Association for Artificial IntelliCopyright gence (www.aaai.org). All rights reserved.
238
GAME THEORY & ECONOMIC MODELS
2. Collect signatures on the contract and make it common
knowledge who signed.
3. Monitor the players’ actions during the execution of G.
4. If the contract was signed by all agents, fine anyone who
deviated from it as specified by the contract.
Our setting is reminiscent of the work on social laws and
conventions (Shoham & Tennenholtz 1997). There too the
center can offer a social convention to the players, where
each player agrees to a particular outcome so long as the
other players play their part. The difference is that in that
work it is assumed that, once all agents agree, the center has
the power to enforce that outcome. Here we assume that
players still have the freedom to choose whether or not to
honor the agreement; the challenge is to design a mechanism
such that, in equilibrium, they will.
The technical results of this paper will refer to games of
complete information, but for intuition consider the example
of online auctions, such as those conducted by eBay. Consider the complete game being played, including the decision after the close of the auction by the seller of whether
to deliver the good and by the buyer of whether to send
payment. Straightforward analysis shows that that the equilibrium is for neither to keep his promise, and the experience with fraud on eBay (Hafner 2004) demonstrates that
the problem is not merely theoretical. It would be in eBay’s
interest to find a way to enable its customers to bind themselves to their promises.
Our first interest will be to characterize the achievable outcomes: What outcomes may the center suggest, with associated penalties, that the agents will accept? Our first result
will be an observation that the center’s power is quite broad:
Any outcome will be accepted when accompanied by appropriate fines, so long as the payoffs of each agent in that
outcome are better than that player’s payoffs in some equilibrium of the original game.
Although the center can achieve almost any outcome, we
note that the helpful center expends a large amount of effort to do so: suggesting an outcome, collecting signatures,
observing the game, and enforcing the contracts. If this procedure happens not just for one game, but for hundreds or
thousands per day, the center may wish to find a way to avoid
this burden while still achieving the same effect.
The bulk of this paper concerns ways in which this reduction in effort can be achieved. We continue to assume that
the center still needs to propose a contract. We also simply
assume that it does not monitor the game. Nor does it participate in the signing phase; the agents do that among themselves using a broadcast channel. While we might imagine
that the players could simply broadcast their signatures, this
protocol allows a single player to learn the others’ signatures
and threaten them with fines. Nonetheless, we can construct
a more complicated protocol - using a second stage of contracts - which does not require the center’s participation. The
only phase in which the center’s protocol requires it to get
involved under some conditions is the enforcement stage.
However, our goal will be to devise contracts so that, in equilibrium, at this stage too the center sits idle. Our results here
will be as follows. If the game play is verifiable (if the center can discover after the fact how the game played out), we
can achieve all of the outcomes achievable by a fully engaged center. If the game is not verifiable then we can still
achieve all previously achievable outcomes with some contract, but that contract might allow equilibria with additional
outcomes.
The rest of the paper is organized as follows: we first formally define our setting. Then we characterize the set of
outcomes which are achievable with a busy center using contracts in this game. Finally, we lighten the load on the center
first in the enforcement stage, then in the signature exchange
stage.
Formal Setting
The strategic situation the center wishes to influence can be
characterized as a strategic-form game with consequences in
O: G = hN, A, O, g, V i. (We roughly follow the notation of
(Osborne & Rubinstein 1994).) Here N is the set of players
{1...n}. A = A1 × A2 × ... × An , where Ai is the set
of actions which can be taken by an individual agent. We
will use ai to refer to an action of i in G and a−i to refer
to the vector of actions of all other players. O is the space
of outcomes; g : A → O determines the outcome after an
action profile. We identify each outcome o(ai ,a−i ) with a
distinct action profile (ai , a−i ), and assume g(ai , a−i ) =
o(ai ,a−i ) . V = V1 × V2 × ... × Vn , where Vi : A → R is the
pay-off function for player i.
Before this strategic situation G occurs, the helpful center
suggests a contract to the players. This contract specifies the
outcome o suggested by the center and what actions h the
center will take in response to different action profiles of the
players. The center will not enforce this contract unless it
is signed by all players. This contract defines the center’s
protocol in the enforcement stage H, as described below.
We will denote a contract that describes a particular center
protocol h as ch .
Now we will describe the stages of the game, as initially
formulated. We will adjust this formulation in later sections
so that the center does no work in equilibrium.
Signature Exchange Stage F Each player who assents
sends his signature on the contract to the center, who collects them. The center notifies all players of the identi-
ties of the signers. At the end of this stage, it is common
knowledge whether or not the contract will be enforced.
Execution Stage G The players play the game G. Each
player may take his action ai to achieve o or he may not.
The center observes the actions taken by the players.
Enforcement Stage H The center takes the actions specified in the contract in response to the actions he observed.
The outcomes are a consequence of the execution stage,
but the only way the center can affect the players’ actions in
the is by fining them in the enforcement stage.
The extended game which arises from playing the stage
games one after another we denote by X = F · G · H.
Together, these define an extensive-form game with simultaneous moves. In general, an extensive-form game X can
be defined as X = hN, Ω, Aω , P, U i, where N is again the
players, Ω is the set of histories of actions taken, Aω is the
set of actions for all players that can be taken after history
ω, P : Ω → 2N is the player function that defines which
players get to move after a given history, and U is the utility
function of players in the entire game.
In our particular setting, the history ω is just the set of
actions taken in each stage game played so far, Aω is the
set of actions possible in each stage game following history
ω, P is the set of all players (all players move simultaneously in each stage), and U is the (undiscounted) sum of
the utilities of each stage game. We denote the subgame of
X = hN, Ω|ω , A|ω , P|ω , U|ω i that arises after history ω by
Γ(ω), which simply refers to the play of the remaining stage
games following the actions taken in ω. In later sections, we
will refer to the strategy space of stage F as ΓF , of stage G
as ΓG , and of stage H as ΓH .
A pure strategy σi for player i in a strategic-form game
corresponds to the choice of a single action σi ∈ Ai . A
mixed strategy corresponds to the choice of a distribution
over actions: σi ∈ ∆Ai . A pure strategy in an extensive
form game is defined as σi : Ω → Aω ; a mixed strategy is
defined accordingly. If σi is a strategy in X, then the strategy σi|ω : Ω|ω → A|ω induced by σi in the subgame Γ(ω)
is σi|ω (ω 0 ) = σi (ω, ω 0 ). Since it is unobservable whether a
player has played a particular mixed strategy (only the realization of that strategy is observed), we will henceforth
concentrate on enforcing outcomes that are the consequence
of pure strategy profiles.
Our chosen solution concept will be subgame perfect
equilibrium. To define this, we must first define a Nash equilibrium: a profile of strategies σ is a Nash equilibrium in a
game if ∀i ∈ N, σi0 ∈ ∆Ai : Ui (σi , σ−i ) ≥ Ui (σi0 , σ−i ).
A profile of strategies σ in an extensive form game is a
subgame perfect equilibrium if for every ω ∈ Ω, σ|ω is a
Nash equilibrium of the subgame Γ(ω). A subgame perfect
equilibrium is resistant to deviations by players even in subgames off the equilibrium path.
The Power of a Helpful Center
We wish to characterize the power of a helpful center without any resource limitations. In this section, the center
is limited only by the voluntary consent required from all
GAME THEORY & ECONOMIC MODELS 239
agents and by its lack of desire to spend its own money.
Specifically, we assume that it collects the signatures in F
itself and that it monitors the players’ actions in G. In later
sections we will relax each of these two assumptions.
First, we must precisely define the game which is being
played. We model the signature exchange stage as a game
form F with players N and action space ΓF = {0, 1}n . The
player i assents to the contract if γiF ∈ ΓF
i = 1. Since the
center broadcasts the identities of the signers, each player’s
action is common knowledge. The execution game G thus
has an extended action space ΓG : {0, 1}n → A in which
players decide to take action based on the consequences of
the signature exchange stage. In the initial formulation, the
enforcement stage H requires no action on the part of the
players, but only of the center. The center’s protocol h sets
the payoff function of the enforcement stage. The center
observes the signatures it receives and the actions chosen by
the players and chooses to fine or reward players. Formally,
h = h1 × h2 × ... × hn and hi : {0, 1}n × O → R.
We define the payoff function Ui : Γ → R for each
player in the extended game X given actions v ∈ {0, 1}n
and (ai , a−i ) ∈ A as Ui (v, ai , a−i ) = V (g(ai , a−i )) +
hi (v, g(ai , a−i )). Thus each player has a quasi-linear utility
function over the outcome determined in G and the money
taken or given by the center according to h.
We say that the center’s protocol h is voluntary if the center neither fines nor rewards players if the contract is not
signed by every player: for all v 6= 1n ∈ {0, 1}n and for all
o ∈ O, it is the case that hi (v, o) = 0. We say that h is frugal
if the center never spends its own money:
for all v ∈ {0, 1}n
P
and all o ∈ O, it is the case that i∈N hi (v, o) ≤ 0. As
these capture the limitations on the helpful center in our setting, we will henceforth limit h to be frugal and voluntary.
We first wish to characterize what outcomes can occur in a
subgame-perfect equilibrium of the extended game X. The
outcome depends on two things: the contract ch suggested
by the center and the strategies of the players in X. We wish
to find contracts to which the players will agree that ensure
that our chosen outcome is played.
In order to characterize the space of possible outcomes
which can be enforced, we must define the notion of a punishment equilibrium. ρi is a punishment equilibrium for i if
ρi is the Nash equilibrium of G with minimal payoffs for i
among all (mixed) Nash equilibria of G.
Theorem 1 Let ρi be the punishment equilibrium for i. For
all o(ai ,a−i ) , if Vi (ai , a−i ) ≥ Vi (ρi ), then there exists a voluntary and frugal center protocol h and a subgame perfect
equilibrium π ∗ in which all players agree to ch and play
(ai , a−i ), and in no subgame perfect equilibrium do players
agree to ch and then fail to play (ai , a−i ). Furthermore, for
all i, Ui (π ∗ ) = Vi (ai , a−i ). If Vi (ai , a−i ) < Vi (ρi ), then
there is no subgame perfect equilibrium in which (ai , a−i )
is played.
Proof: First, suppose Vi (ai , a−i ) < Vi (ρi ). Since player
i will get at least Vi (ρi ) in any subgame perfect equilibrium without fines, i can profit by withholding his assent.
As (ai , a−i ) cannot be a Nash equilibrium by assumption
and no fines are assessed in H, there can be no subgame
240
GAME THEORY & ECONOMIC MODELS
perfect equilibrium in which (ai , a−i ) is played.
Second, suppose Vi (ai , a−i ) ≥ Vi (ρi ). We choose
hi (ai ) = 0 and hi (a0i 6= ai ) = −M . If we choose
M so that for all i, a0i , and a0−i , it is the case that M >
Vi (a0i , a0−i )−Vi (ai , a−i ), then (ai , a−i ) will be the only subgame perfect equilibrium of the subgame G · H, supposing
all players agree to ch . We also require that all players assent
in F . If any player does not assent, all players coordinate on
his punishment equilibrium ρi in G. If more than one player
fails to assent, we break ties arbitrarily to see which ρi is
played. No matter which player fails to assent, ρi will be a
subgame perfect equilibrium of G · H, since the center will
not assess fines. Thus i will not profit by withholding his
assent.
Thus we show that, with a fully engaged center that takes
part in the protocol and monitors the players’ actions, we
can achieve any payoffs for the players which are at least
as good for every player as some Nash equilibrium of G.
Furthermore, once a contract for o is mutually signed, the
unique subgame perfect equilibrium achieves o.
We notice that, already, the center takes no action in H
in equilibrium. Yet as the center takes action in every other
stage, we shall consider how to lighten the load on the center.
Removing the Center From the Enforcement
Stage
In this section, we will drop the assumption that the center
does not monitor the players’ actions in the execution stage
G. Instead, we assume that actions and outcomes are common knowledge among the players but are not observed by
the center. The center must therefore encourage the players
to tell him if there has been a deviation. We will distinguish
two cases. In the verifiable case, the center can verify that
a particular player played a given action if he chooses to
do so once the game G has been played. Specifically, we
require that the center be able to verify, for each player i,
whether i played the correct action ai or some other action
a0i 6= ai . The center saves effort by not paying attention to
G; we merely require that he can determine the truth after
the fact, if necessary. In the unverifiable case, the center has
no information about players’ actions whatsoever.
Because we now require the center to be notified by the
players of deviations, the enforcement games we now consider will be of the following form: first, the players observe
the outcome and send messages to the center. The center
publishes any messages he receives to all players. The players then have the chance to respond to the center’s messages.
This repeats for some number of rounds. Finally, the center
makes monetary transfers between the players based on the
messages sent.
For our purposes, this full generality is not needed. Our
enforcement stage H is a single-round stage game where
each player chooses whether or not to complain about
other players by sending their names to the center, and
the center chooses a fine to impose on each player: H =
N
specifies which
hN, ΓH , Rn , hi. γiH ∈ ΓH
n : O → 2
complaints player i will send to the center after each outcome. As before, h is the center’s protocol which maps out-
comes and complaints received to monetary consequences
in Rn . The center may make payments based on the outcome (if he can verify it), the identities of the complainers,
and the target of their complaints. In the verifiable case,
h : O × (2N )n → Rn , while in the unverifiable case,
h : (2N )n → Rn .
Now that we have specified an enforcement game, we
wish to characterize the set of outcomes obtainable thereby
in the extended game corresponding to this enforcement
game.
We define a protocol ho for the center, which will induce
an equilibrium under which the center takes no action in
the enforcement stage. Let M and m be a large and small
amount of money, respectively. In ho , the center punishes
each player who deviated by a large-enough amount M , but
also rewards each player who sent in a correct complaint by
m for each correct complaint. The center also punishes any
player who sent in an incorrect complaint by m. The contract that specifies center protocol ho we call cho .
Theorem 2 (Contracts for Verifiable Games) Let G be a
game with verifiable consequences in O and let o(ai ,a−i ) ∈
O be the desired outcome. Assume that the center has suggested contract cho defined above and consider the subgame
G · H that follows unanimous agreement to this contract.
Then there is a strategy profile π ∗ such that π ∗ is the unique
subgame perfect equilibrium of G · H, o(ai ,a−i ) is the equilibrium outcome of π ∗ , and π ∗ has payoffs V (ai , a−i ). The
center takes no action if π ∗ is played.
[Proof omitted.]
We now consider the unverifiable case. As before, we first
define a particular center protocol h0o . In h0o , the center punishes the target of each complaint by a large-enough amount
M , but does not reward or punish players for complaints.
After all, the center cannot distinguish valid complaints from
invalid ones. The contract that specifies center protocol h0o
we call ch0o .
Theorem 3 (Contracts for Unverifiable Games) Let G be
a game with unverifiable consequences in O, and let
o(ai ,a−i ) ∈ O be the desired outcome. Assume that the center has suggested contract ch0o and consider the subgame
G · H that follows unanimous agreement to this contract.
Then there is a strategy profile π ∗0 such that π ∗0 is a subgame perfect equilibrium of G · H, o(ai ,a−i ) is the equilibrium outcome of π ∗0 , and π ∗0 has payoffs V (ai , a−i ). The
center takes no action if π ∗0 is played.
[Proof omitted.]
We have seen that even without verifiability, it is possible to achieve almost any outcome in equilibrium. Unfortunately, these equilibria are no longer unique. As we shall
see, in the unverifiable case, a given signed contract may
have many possible equilibrium outcomes rather than just
the intended one.
Given a game G, define the shortfall sσi of pure-strategy
profile σ = (ai , a−i ) for i as sσi = maxa0i Vi (a0i , a−i ) −
Vi (ai , a−i ). The shortfall of i in σ is the amount i’s payoffs would need to rise so that i would have no incentive
to deviate from σ, all else held constant. We can see that
there must be some equilibrium of the enforcement game
(a ,a )
in which an agent i is punished by at least si i −i whenever he deviates from his action ai . Yet, in an unverifiable
game, there is nothing in the center’s protocol which makes
(ai , a−i ) special. The players could just as well coordinate
on this equilibrium in the enforcement game when the actions are not some other action (a0i , a−i ). This implies that
any enforcement scheme for the unverifiable case will not in
general have a unique outcome. Here we consider not only
our chosen center protocol h0o , but in fact any center protocol
h in any form of enforcement game H.
Theorem 4 (Spurious Equilibria) Consider an unverifiable enforcement game with a frugal and voluntary h under
which G · H has a subgame-perfect equilibrium π in which
the center does no work, where (ai , a−i ) is the strategy profile that π plays in G. Then if σ 0 is a pure strategy profile
0
(a ,a )
of G and ∀i : sσi ≤ si i −i , then there exists a subgame
0
perfect equilibrium π of G · H such that σ 0 is the strategy
profile that π 0 plays in G.
[Proof omitted.]
A consequence of this theorem is that any Nash equilibrium of G can be played in F ·G·H regardless of the contract
signed.
Corollary 5 (No Deletion) If σ is a pure or mixed Nash
equilibrium in the unverifiable game G, then, for any frugal and voluntary center protocol h that has a subgame perfect equilibrium π where the center does no work, there is a
subgame perfect equilibrium π 0 of G · H such that σ is the
strategy profile that π 0 plays in G.
[Proof omitted.]
Thus, if the center cannot verify the players’ actions, he
cannot in general enforce any outcomes uniquely. After
signing the contracts, the players might arrive at an outcome
different from the one the center suggested. In a real-world
setting, this would substantially weaken the case that the
players should sign the contract.
We have shown that a helpful center who neither monitors the player’s actions nor fines any player in equilibrium
can enforce every outcome that a fully engaged center can
enforce with more burdensome contracts. In an unverifiable
game, however, the center must generally accept spurious
equilibria. Our next task is to remove the center from the
signature exchange stage.
Exchanging Signatures Without The Center
Under the original contract, the center collected signatures
on the contract ch and enforced the contract if every player
signed. We now show how the players can exchange signatures on the contract by use of a broadcast channel without
requiring any action from the center in equilibrium. In this,
our goal is similar to the goal of optimistic signature exchange (Garay & MacKenzie 1999), but with rational actors
instead of computationally-bounded ones.
If players may communicate without being observed by
others, F would be a game of imperfect information. As
these games are difficult to analyze and generally admit of
GAME THEORY & ECONOMIC MODELS 241
many solutions, we require the players to use a broadcast
channel, on which all messages sent are common knowledge.
When the center no longer monitors the signature exchange stage, he no longer knows in the enforcement stage
H whether the contracts have been signed or not. Therefore, we now require that each complaint sent to the center
in the enforcement stage H include a fully signed copy of
the contract.
The Naive Broadcast Protocol
We might hope that the signature collection service performed by the center was superfluous: that we will achieve
the same results if we simply require players to broadcast their agreement or disagreement. Unfortunately, this
will not be so. Consider the naive broadcast protocol
where all players simultaneously broadcast their signatures.
Let us formally define F to be the one-round stage game
F = hN, ΓF , S, f i, where N is the set of players and
ΓF = {0, 1}n, where 0 represents the decision not to broadcast one’s signature, while 1 represents the decision to do
so. S = ({0, 1}n)n is the set of outcomes of the game.
Each outcome specifies the set of signatures (represented by
{0, 1}n) possessed by each player in N . f : ΓF → S is the
outcome function of F : each player knows his own signature and every signature which is broadcast.
The complete set of signatures is thus common knowledge
if and only if every player chooses to broadcast his signature.
Consider what occurs if exactly one player i fails to reveal
his signature: i has received all the signatures of the other
players, and he can produce his own. Thus i is the only
player to possess all signatures on the contract, and this fact
is common knowledge among the players. The center, on
the other hand, cannot distinguish this case from the case
where all players know all signatures, but only i chooses
to complain. Therefore i is able to unilaterally enforce the
contract, unlike in the original formulation.
Recall that, in H, every message sent by one player to
the center is broadcast to all other players. Thus, once one
player has sent a complaint about another (which includes a
fully signed contract), every player will know all signatures
on ch and be able to complain. A player who deviates in F
cannot choose to punish other players while remaining unscathed himself, but he can choose unilaterally whether or
not to enforce the contract. Unfortunately, this power implies that our previously specified equilibrium for the extended game F · G · H is no longer an equilibrium.
The equilibrium for F ·G·H discussed above requires that,
if i fails to reveal his signature on the contract, all players coordinate on i’s punishment equilibrium. Consider the case,
for instance, where the punishment equilibrium for some i
is (ai , a0−i ), where ai is the action i is contractually obliged
to play. Suppose i alone fails to reveal his signature and all
players play i’s punishment equilibrium (ai , a0−i ). In stage
H, then, i will profit by choosing to enforce the contract: the
center will punish the other players and reward i. Knowing
this, the other players will not in general wish to play their
part of the punishment equilibrium, so our previous strategy
fails.
242
GAME THEORY & ECONOMIC MODELS
The Pre-Contract Protocol
Although the naive broadcast protocol did not allow us to
guarantee all the payoffs we wanted, we shall see that we
can use a more complicated signature exchange stage F to
ensure that either each player receives all signatures on ch ,
or no players receive all signatures on ch . Our exchange
scheme is modelled on the contracts mechanism of the rest
of the paper: we will add a pre-contract ĉh that the players will sign before signing ch . This contract authorizes the
center to fine players who do not reveal their signature on ch .
Surprisingly, this does not lead to infinite regress: this one
pre-contract is sufficient to allow for signature exchange.
We will divide F itself into stages: a miniature contract
exchange stage F̂ , a miniature execution stage Ĝ, and a
miniature enforcement stage Ĥ. The players will bind themselves in contract ĉh to reveal their signatures on the contract
ch in such a way that, if they fail to reveal them, they can be
fined by the center. We will allow them, however, to recoup
that fine by revealing their signatures on ch to the center and
all players after the fact. The after-the-fact alteration of the
outcome allows us to use the naive broadcast protocol for F̂
where we could not use it for F .
Formally, let the signature stage F = F̂ · Ĝ · Ĥ. Let
us call the contract signed in F̂ the pre-contract ĉh , which
binds players to release their signatures on the real contract
ch . F̂ is the naive broadcast protocol defined above as the
stage game F̂ = hN, ΓF̂ , S, fˆi. S is the set of signatures on
ĉh that each player knows, and ΓF̂ is each player’s choice to
broadcast or not broadcast his signature on ĉh . Ĝ is also the
naive broadcast protocol defined above for the signatures on
ch : Ĝ = hN, ΓĜ , Φ, ĝi, with Φ the set of known signatures
on ch , and ΓĜ : S → 0, 1n the decision of the players to
broadcast their signatures on ch given what signatures each
player knows on ĉh .
Ĥ = hN, ΓĤ , Rn , ĥi is a miniature enforcement stage
that is substantially different from the enforcement stage H.
Ĥ has two rounds. In the first round, a player i is allowed
to complain to the center that he has not received some signature on ch . To do so, i must submit the contract ĉh , all
signatures on ĉh , and i’s signature on ch . When the center
rebroadcasts this message to all players, both the signatures
on ĉh and i’s signature on ch become known to all players. In the second round, each player who did not complain
in the first round is given a chance to complain. ΓĤ
i simply characterizes whether i will complain to the center after
each history.
We now state our chosen center protocol ĥ in Ĥ. If the
center received a complaint in the first round, then the center fines all players who did not complain in either the first
or the second round by a large-enough amount M . If the
center does not receive any complaints, he does not fine any
players. Note that if all players complain, all signatures on
ch become common knowledge and no fines are assessed.
We now specify the strategy π ∗ we expect the agents to
play in F . In F̂ , each player first reveals his signature on
ĉh . In Ĝ, he will reveal his signature on ch if and only if
he has received the signatures of every other player on ĉh .
In the enforcement stage Ĥ, he complains to the center if
and only if he has received all signatures on ĉh , but he has
not received all signatures on ch , or if some other player has
complained.
The remainder of π ∗ for G and H is simple. If all signatures on ch become known to all players, then the players
play (ai , a−i ) in G to achieve o, just as before, and then
complain to the center in H if some player deviates. If,
however, some player i deviates from the equilibrium in F
(whether by choosing not to reveal in F̂ , choosing not to
reveal in Ĝ, or failing to complain in Ĥ), in such a way
that the signatures on ch do not become commonly known,
then the agents coordinate on that player’s punishment equilibrium ρi . If several players deviate during F , the agents
coordinate on the punishment equilibrium of the last player
to deviate.
Our strategy π ∗ is now an equilibrium. Thus, we can
achieve any outcome achievable with a busy center with a
center that does no work in equilibrium.
Theorem 6 Let X be the extended game F̂ · Ĝ · Ĥ · G · H,
and let ρi be the punishment equilibrium for i in G. Then,
for any o(ai ,a−i ) such that for all i, Vi (ai , a−i ) ≥ Vi (ρi ),
there exists a contract ch for which there is an strategy profile π ∗ such that π ∗ is a subgame perfect equilibrium of the
extended game and o(ai ,a−i ) is the equilibrium outcome of
π ∗ . Furthermore, in π ∗ , the center takes no action during
any stage.
Proof: Let us sketch why this will be a subgame perfect
equilibrium. We will proceed by backwards induction.
So long as no single player gains complete knowledge of
the signatures on ch , then π ∗ is a subgame perfect equilibrium in G and H. This does not occur if π ∗ is played in F ,
so it is sufficient to prove that π ∗ is a subgame perfect equilibrium in F . We will show that, even after one deviation,
either all players know all the signatures on ch or no player
knows all signatures on ch .
Consider the second round of Ĥ. If no player complained
in the first round, second-round complaints have no effect.
If a player complained in the first round of Ĥ, then it will
be dominant for every other player to complain according to
π ∗ to avoid the punishment of M from the center. Thus, all
players will know all signatures on ch .
Consider the first round of Ĥ. There are three cases to
distinguish. First, if every player knows all signatures on
both ĉh and ch , then complaining will have no effect. Second, if every player knows all signatures on ĉh , but only one
player knows all signatures on ch . Every other player will
complain in the first round and i must therefore complain in
the first or second rounds to avoid losing M . Every player
will learn all signatures. Third, if only one player i knows all
the signatures on ĉh , then i will not know all the signatures
on ch . If i does not complain, no player will learn all signatures on ch and players will coordinate on i’s punishment
equilibrium. If i does complain, all others will complain in
the second round and all players will learn all signatures.
Consider the stage Ĝ. There are now two cases to consider. First, suppose all players know all signatures on ĉh .
Then no player i can benefit by failing to reveal his signature
on ch , since the other players will complain, i will complain
to avoid punishment, and all players will end up learning all
signatures on ch . Second, suppose only one player i knows
the signatures on ĉh because he failed to reveal in F̂ . Then
no other players will reveal their signatures, and, whether i
reveals or not, all players will coordinate on i’s punishment
equilibrium.
Finally, consider the stage F̂ . Suppose one player i deviates in the stage F̂ by failing to reveal his signature on the
pre-contract ĉh . Then he alone will have all the signatures
on ĉh , and no one else will reveal their signatures on ch in Ĝ.
According to the equilibrium, i will complain to the center
in stage Ĥ, resulting in complete knowledge of ch and the
decision to play (ai , a−i ).
Conclusion
We have discussed the power of a helpful center in enabling
a group of players to make contracts which require them to
play a certain strategy or face penalties. Even if the center
brings no money to the system and transfers money from the
players only after receiving permission, the center is able to
help the players achieve nearly any outcome of the game.
Moreover, we find that the center is still able to help the
players achieve these outcomes in equilibrium, even if he
does not monitor the game and does not participate on the
equilibrium path - in other words, even when the center does
no work in equilibrium beyond suggesting a contract.
In fact, if the contracts the center would suggest are common knowledge or determined by a negotiation stage between the agents, the center does no work whatsoever in
equilibrium. Incidentally, we notice that the center makes
a profit to cover his costs whenever his services are used.
These two properties are very important for a third party who
wishes to influence outcomes in strategic settings that occur
frequently, such as in the online auction setting.
References
Garay, J. A., and MacKenzie, P. D. 1999. Abuse-free multiparty contract signing. In International Symposium on Distributed Computing, 151–165.
Hafner, K. 2004. Vigilantes attack eBay fraud. The
New York Times. Available at http://news.com.com/21001038 3-5176525.html.
Mas-Colell, A.; Whinston, M.; and Green, J. 1995. “Microeconomic Theory”. Oxford University Press.
Monderer, D., and Tennenholtz, M.
2003.
kimplementation. In Proceedings of the 4th ACM Conference on Electronic Commerce, 19–28.
Osborne, M., and Rubinstein, A. 1994. A Course in Game
Theory. MIT Press.
Shoham, Y., and Tennenholtz, M. 1997. On the emergence
of social conventions: Modeling, analysis, and simulations.
Artificial Intelligence 94:139–166.
GAME THEORY & ECONOMIC MODELS 243